134 research outputs found
On the complexity of solving ordinary differential equations in terms of Puiseux series
We prove that the binary complexity of solving ordinary polynomial
differential equations in terms of Puiseux series is single exponential in the
number of terms in the series. Such a bound was given by Grigoriev [10] for
Riccatti differential polynomials associated to ordinary linear differential
operators. In this paper, we get the same bound for arbitrary differential
polynomials. The algorithm is based on a differential version of the
Newton-Puiseux procedure for algebraic equations
An Algorithm for Computing the Limit Points of the Quasi-component of a Regular Chain
For a regular chain , we propose an algorithm which computes the
(non-trivial) limit points of the quasi-component of , that is, the set
. Our procedure relies on Puiseux series expansions
and does not require to compute a system of generators of the saturated ideal
of . We focus on the case where this saturated ideal has dimension one and
we discuss extensions of this work in higher dimensions. We provide
experimental results illustrating the benefits of our algorithms
Analogue of Newton-Puiseux series for non-holonomic D-modules and factoring
We introduce a concept of a fractional-derivatives series and prove that any
linear partial differential equation in two independent variables has a
fractional-derivatives series solution with coefficients from a differentially
closed field of zero characteristic. The obtained results are extended from a
single equation to -modules having infinite-dimensional space of solutions
(i. e. non-holonomic -modules). As applications we design algorithms for
treating first-order factors of a linear partial differential operator, in
particular for finding all (right or left) first-order factors
Computing Puiseux series : a fast divide and conquer algorithm
Let be a polynomial of total degree defined over
a perfect field of characteristic zero or greater than .
Assuming separable with respect to , we provide an algorithm that
computes the singular parts of all Puiseux series of above in less
than operations in , where
is the valuation of the resultant of and its partial derivative with
respect to . To this aim, we use a divide and conquer strategy and replace
univariate factorization by dynamic evaluation. As a first main corollary, we
compute the irreducible factors of in up to an
arbitrary precision with arithmetic
operations. As a second main corollary, we compute the genus of the plane curve
defined by with arithmetic operations and, if
, with bit operations
using a probabilistic algorithm, where is the logarithmic heigth of .Comment: 27 pages, 2 figure
Singular Points of Real Quartic and Quintic Curves
There are thirteen types of singular points for irreducible real quartic curves and seventeen types of singular points for reducible real quartic curves. This classification is originally due to D. A. Gudkov. There are nine types of singular points for irreducible complex quartic curves and ten types of singular points for reducible complex quartic curves. There are 42 types of real singular points for irreducible real quintic curves and 49 types of real singular points for irreducible real quintic curves. The classification of real singular points for irreducible real quintic curves is originally due to Golubina and Tai. There are 28 types of singular points for irreducible complex quintic curves and 33 types of singular points for reducible complex quintic curves. We derive the complete classification with proof by using the computer algebra system Maple. We clarify that the classification is based on computing just enough of the Puiseux expansion to separate the branches. Thus, the proof consists of a sequence of large symbolic computations that cam be done nicely using Maple
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